U.S. patent number 4,555,767 [Application Number 06/442,193] was granted by the patent office on 1985-11-26 for method and apparatus for measuring thickness of epitaxial layer by infrared reflectance.
This patent grant is currently assigned to International Business Machines Corporation. Invention is credited to William R. Case, Wildey E. Johnson.
United States Patent |
4,555,767 |
Case , et al. |
November 26, 1985 |
Method and apparatus for measuring thickness of epitaxial layer by
infrared reflectance
Abstract
Method and apparatus measure the thickness of an epi layer grown
on a substrate. IR energy 12 is directed onto the epi layer 13 and
a portion 14 of the energy is reflected from the surface of the epi
layer and from the interface of the epi layer and substrate. The
spectral reflectance of the reflected energy is measured by means
of a Fourier transform IR spectrometer 10. The measured values of
spectral reflectance are correlated with a series of theoretical
reflectance values determined for different thicknesses of an epi
layer in a range including the nominal thickness. The measured or
actual epi thickness is determined from the correlation
analysis.
Inventors: |
Case; William R. (Walden,
NY), Johnson; Wildey E. (Boca Raton, FL) |
Assignee: |
International Business Machines
Corporation (Armonk, NY)
|
Family
ID: |
26766528 |
Appl.
No.: |
06/442,193 |
Filed: |
October 28, 1982 |
PCT
Filed: |
May 27, 1982 |
PCT No.: |
PCT/US82/00729 |
371
Date: |
October 28, 1982 |
102(e)
Date: |
October 28, 1982 |
Current U.S.
Class: |
250/341.4;
250/358.1; 356/504; 356/632 |
Current CPC
Class: |
G01B
11/0625 (20130101) |
Current International
Class: |
G01B
11/06 (20060101); G01B 011/02 () |
Field of
Search: |
;364/563,728
;356/355,357,381,382 ;156/601 ;148/175 |
References Cited
[Referenced By]
U.S. Patent Documents
Other References
IBM Technical Disclosure Bulletin, Jun. 1981 (vol. 24, No. 1A), W.
R. Case; "Transparent Film Thickness Measurement", pp. 49-54. .
IBM Technical Disclosure Bulletin, Jun. 1982 (vol. 25, No. 1), G.
H. Hewig; "In-Situ, Realtime Thin-Film Refractive Index &
Thickness Monitor", pp. 436-438. .
IBM Technical Disclosure Bulletin, Aug. 1976 (vol. 19, No. 3), R.
B. Ananthakrishan, "Cubic Spline Fit for Calibration of Thin Film
Measurement Tools", pp. 890-896. .
Applied Optics, Sep. 1978 (vol. 17, No. 17), A. M. Goodman;
"Optical Interference Method for Approximate Determination of
Refractive Index and Thickness of a Transparent Layer", pp.
2779-2787. .
IBM Technical Disclosure Bulletin, Apr. 1976 (vol. 18, No. 11), R.
B. Ananthakrishnan; "Algorithm for Computing Thin-Film
Thicknesses", pp. 3618-3819. .
J. Electrochem. Soc., Feb. 1973 (vol. 120, No. 2), P. F. Cox et
al.; "Measurement of Si Epitaxial Thickness Using a Michelson
Interferometer", pp. 287-292. .
J. App. Phy., Jul. 1970 (vol. 41, No. 8), P. A. Schumann et al.,
"Measurement of Silicon Epitaxial Layers Less than 1.mu. Thick by
Infrared Interference", pp. 3532-3535. .
"Optical Properties of Thin Solid Films"; O. S. Heavens, Phd,
published by Butterworths Scientific Publications, 1955, pp. 76 and
77..
|
Primary Examiner: Krass; Errol A.
Assistant Examiner: Teska; Kevin J.
Attorney, Agent or Firm: McKechnie; Douglas R.
Claims
Having thus described by invention, what we claim as new, and
desire to secure by Letters Patent is:
1. The method of measuring, by means of an instrument connected to
a data processing system (DPS), the thickness of a sample epi layer
grown on a substrate, comprising the steps of:
(a) operating said instrument and said DPS to measure the spectral
reflectance of IR energy reflected from the surface of the epi
layer and from the interface between the epi layer and the
substrate;
(b) storing within said DPS values representing the measured
spectral reflectance resulting from step (a);
(c) storing within said DPS values representing a series of
theoretical spectral reflectances determined for a series of epi
layers of different thicknesses within a range including the
thickness of the epi layer being measured, said series of epi
layers being of like doping concentrations;
(d) correlating within said DPS values for each of said theoretical
spectral reflectances with values of said measured spectral
reflectance and providing a quantitative measure of the degree of
correlation as a function of epi layer thickness;
and (e) determining within said DPS the thickness of the sample epi
layer from the maximum value of said quantitative measure of the
degree of correlation.
2. The method of claim 1 wherein said theoretical spectral
reflectance values is calculated on the basis of the assumption
that the epi layer on the substrate is an absorbing thin layer on
an absorbing substrate.
3. The method of claim 2 wherein said theoretical spectral
reflectance values are calculated using equations eqn. 1 through
eqn. 17.
4. The method of claim 1 wherein step (a) comprises the sub-steps
of
(a1) measuring the intensity of IR energy reflected from the layer
and the interface, by means of an interferometer,
(a2) and Fourier transforming, within said DPS, the measured
intensity to produce said spectral reflectance.
5. The method of claim 1 wherein:
step (d) comprises calculating a series of correlation coefficients
as a function of epi thickness;
and step (e) comprises fitting a quadratic curve to said
correlation coefficients and determining the thickness of the
sample from the maximum positive peak of said curve.
6. Apparatus for measuring the thickness of a sample epi layer
grown on a substrate, comprising:
measuring means for measuring the spectral reflectance of IR energy
reflected from the epi layer and the interface between the epi
layer and the substrate;
and data processing means for performing plural functions
including
storing digital values representing the measured spectral
reflectance,
storing digital values representing a series of theoretical
spectral reflectances determined for a series of epi layers of
different thicknesses within a range encompassing the thickness of
the sample epi layer, said series of epi layers being of like
doping concentration,
correlating each theoretical spectral reflectance with said
measured spectral reflectance and providing a series of signals
representing quantitative measurements of the degree of correlation
there between as a function of epi layer thickness,
and determining the thickness of the sample epi layer from the
maximum value of said quantitative measurement.
7. Apparatus of claim 6 wherein said measuring means comprises:
a Fourier transform IR spectrometer for measuring the intensity of
the reflected IR energy and converting such intensity into said
spectral reflectance.
8. Apparatus of claim 6 wherein:
said data processing means performs the function of for calculating
said theoretical spectral reflectances on the assumption that the
epi layer on the substrate forms an absorbing layer on an absorbing
substrate.
9. Apparatus of claim 6 wherein said data processing means
calculates a series of correlation coefficients, and determines the
epi thickness from said correlation coefficients.
10. Apparatus of claim 9 wherein said data processing means
calculates said series of correlation coefficients using equation
eqn. 18.
11. The method of measuring, by means of an instrument connected to
a data processing system (DPS), the thickness of a sample epi layer
grown on a substrate, comprising the steps of:
(a) operating said instrument and said DPS to measure the spectral
reflectance of IR energy reflected from the surface of the epi
layer and from the interface between the epi layer and the
substrate;
(b) storing within said DPS values representing the measured
spectral reflectance resulting from step (a);
(c) storing within said DPS values representing a series of
theoretical spectral reflectances determined for a series of epi
layers of different thicknesses within a range including the
thickness of the epi layer being measured, said series of epi
layers being of like doping concentrations;
(d) correlating within said DPS values of each of said theoretical
spectral reflectances with values of said measured spectral
reflectance and providing a quantitative measure of the degree of
correlation as a function of epi layer thickness, said quantitative
measure being calculated using eqn. 18;
and (e) determining within said DPS the thickness of the sample epi
layer from the quantitative measure.
12. The method of measuring epi layer thicknesses within a
sub-micron range, by means of an instrument connected to a data
processing system (DPS), the thickness of a sample epi layer grown
on a substrate, comprising the steps of:
(a) operating said instrument and said DPS to measure the spectral
reflectance of IR energy reflected from the surface of the epi
layer and from the interface between the epi layer and the
substrate;
(b) storing within said DPS values representing the measured
spectral reflectance resulting from step (a);
(c) storing within said DPS values representing a series of
theoretical spectral reflectances determined for a series of epi
layers of different thicknesses within a range including the
thickness of the epi layer being measured, said series of epi
layers being of like doping concentrations;
(d) correlating within said DPS values of each of said theoretical
spectral reflectances with values of said measured spectral
reflectance and providing a quantitative measure of the degree of
correlation as a function of epi layer thickness;
and (e) determining within said DPS the thickness of the sample epi
layer from the quantitative measure, said thickness being in a
range of thicknesses comprising values from less than 0.1 micron to
1.0 micron.
13. Apparatus for measuring the thickness of a sample epi layer
grown on a substrate, comprising:
measuring means for measuring the spectral reflectance of IR energy
reflected from the epi layer and the interface between the epi
layer and the substrate;
and data processing means for performing plural functions
including
storing digital values representing the measured spectral
reflectance,
calculating and storing digital values representing a series of
theoretical spectral reflectances determined for a series of epi
layers of different thicknesses within a range encompassing the
thickness of the sample epi layer, said series of epi layers being
of like doping concentration, said calculating being done using
equations eqn. 1 through eqn. 17 and using the assumption that the
epi layer on the substrate forms an absorbing layer on an absorbing
substrate,
correlating each theoretical spectral reflectance with said
measured spectral reflectance and providing a series of signals
representing quantitative measurements of the degree of correlation
therebetween as a function of epi layer thickness,
and determining the thickness of the sample epi layer from said
quantitative measurements.
14. Apparatus for measuring the thickness of a sample epi layer
grown on a substrate, comprising:
measuring means for measuring the spectral reflectance of IR energy
reflected from the epi layer and the interface between the epi
layer and the substrate;
and data processing means operable to determine epi layer thickness
over the range from less than 0.1 micron to greater than 35
microns, said data processing means performing plural functions
including
storing digital values representing the measured spectral
reflectance,
storing digital values representing a series of theoretical
spectral reflectances determined for a series of epi layers of
different thicknesses within a range encompassing the thickness of
the sample epi layer, said series of epi layers being of like
doping concentration,
correlating each theoretical spectral reflection with said measured
spectral reflectance and providing a series of signals representing
quantitative measurements of the degree of correlation therebetween
as a function of epi layer thickness,
and determining the thickness of the sample epi layer from said
quantitative measurements.
Description
FIELD OF THE INVENTION
This invention relates to improvements in method and apparatus for
measuring or determining the thickness of an epitaxial (epi) layer
on a semiconductor wafer by use of an infrared (IR) reflectance
technique.
BACKGROUND OF THE INVENTION
As is well known, it is important in the fabrication of
semi-conductor devices to know the thickness of an epi layer on a
semi-conductor wafer. Different methods are known within the prior
art for measuring or determining the epi thickness including
methods based upon IR interference physical optic theory. In
accordance with such theory, IR energy is directed onto a wafer and
is reflected from the surface of the epi layer and from the
interface between the epi layer and underlying substrate. The IR
energy is directed as an incident beam onto a small area of the
wafer at a position where the epi thickness is to be measured. Such
incident beam is divided to form two reflected beams. One beam is
reflected from the surface of the epi layer and the other beam is
reflected from the epi layer/substrate interface. The two reflected
beams interface with each other in such a manner that the epi
thickness can be determined by spectral reflectance and
interferometric methods.
The spectral reflectance method is based on the phenomena that the
degree of optical interference between the two reflected beams
cyclically varies at each wavelength across a spectrum. The
variation produces a series of maxima and minima reflectance values
in accordance with the degree of constructive and destructive
interference at the different wavelengths. Such method generally
involves measuring the spectral reflectance and then calculating
the thickness using the reflectance at two different maxima or
minima. An example of this method is standard test method F95 of
the American Society for Testing Materials (ASTM) for "Thickness of
Epitaxial Layers of Silicon on Substrates of the Same Type by
Infrared Reflectance".
In the interferometric method, an interferometer is used to
generate an interferogram from the two reflected beams. The
interferogram includes a center burst or peak and two side bursts
or peaks created as a result of displacement of the interferometer
mirror. In a perfect system, the interferometer would be symetrical
and the degree of mirror displacement between two positions
corresponding to two of the bursts or peaks, is proportional to the
epi thickness. In actual practice however, the interferogram is
asymmetrical and a double Fourier Transform and other mathematical
manipulations are performed to create an idealized interferogram,
from which the thickness is calculated as a function of mirror
displacement between the side peaks. An example of this method is
described in "Measurement of Si Epitaxial Thickness Using a
Michaelson Interferometer", Paul F. Cox and Arnold F. Stalder, J.
Elec. Soc.: Solid State Science and Technology, February 1973, pgs.
287-292.
A current trend in the semi-conductor industry is to grow thinner
and thinner epi layers having thicknesses less than one-half a
micron. Thus there has developed the need to measure the
thicknesses of such thin layers. But the methods and apparatuses of
the prior art, particularily commercially available instruments,
appear to be limited to making accurate measurements only for
thicknesses substantially above 0.5 microns. In the spectral
reflectance method, the number of extrema decrease with a decrease
in epi thickness and it becomes difficult or impossible to average
a number of calculations per the ASTM method, or even to recognize
the extrema. In the interferometric method, the side peaks
interfere with the center peak below three microns and with each
other at thinner thicknesses, so that the mirror displacement
cannot be accurately determined.
Additionally, prior art theory appears to be based on a number of
simplifying assumptions about some of the optical factors or
conditions, which assumptions are not exactly true and which tend
to produce inaccurate results when applied to the measurement of
extremely thin epi layers. Examples of such assumptions are that
the index of refraction is constant, that there is no phase shift
at the epi layer/substrate interface, and that the substrate is
non-absorbant.
SUMMARY OF THE INVENTION
One of the objects of the invention is to provide an improved
method and apparatus for rapidly and accurately measuring the
thickness of an epi layer within a range the lower limit of which
is less than 0.1 micron.
Another object is to determine the epi thickness from spectral
reflectance data, in an improved manner.
Still another object is to measure epi thickness in accordance with
physical optic theory without having to calibrate any apparatus or
method.
Briefly, in accordance with the invention, a wafer having an epi
layer grown on a substrate to a nominal thickness, is irradiated
with IR energy and the spectral reflectance is measured. The
measured values are correlated with a series of theoretical values
determined for different thicknesses of an epi layer in a range
including the nominal thickness. The actual epi thickness is
determined from the correlation analysis. The theoretical values
are based on the assumption of solid thin film theory that the epi
layer constitutes a single absorbing layer on an absorbing
substrate, and that the index of refraction and absorption
coefficients vary.
Other objects and advantages of the invention will be apparent from
the following description of a preferred embodiment taken in
connection with the accompanying drawings wherein:
FIG. 1 is a block diagram of apparatus embodying the invention;
FIG. 2 is a flow diagram of the method of the invention;
FIG. 3 is a diagrammatic illustration useful in understanding the
invention;
FIG. 4 is a spectral reflectance graph derived from measuring a
sample;
FIG. 5 is a correlation graph derived from the same sample from
which the graph of FIG. 4 was derived;
FIG. 6 is a spectral reflectance graph derived from another sample;
and
FIG. 7 is a correlation graph derived from the sample from which
the graph from FIG. 6 was derived.
DETAILED DESCRIPTION
In general, the apparatus shown in FIG. 1 comprises a known,
commercially available Fourier transform infrared (FTIR)
spectrometer 10, such as one of the models of the IR80 or 90 series
of FTIR spectrometers marketed by IBM Instruments Inc., modified
(1) by using a reflectance accessory 11 to direct incident IR
energy 12 onto a semiconductor wafer 13 and collect IR energy 14
reflected from the wafer, and (2) by the addition of novel programs
stored in a data processing system (DPS) 15 for controlling
operation of the spectrometer to measure or determine the thickness
of an epi layer grown on the substrate of a wafer. Since most of
the apparatus seen in FIG. 1 is old and well known, only a general
description thereof is necessary in order to understand the
invention.
An IR source 16 produces a beam of IR energy in the mid-IR region
in which the epi layer is transparent. The beam from source 16 is
directed into an interferometer 17 which includes a conventional
beam splitter and movable mirror (not shown). These components
function to split the beam into two beams and later re-combine the
two beams with a degree of interference dependent upon the
difference in optical path lengths of the two split beams. The
degree of optical path length difference is controlled by the
displacement of the movable mirror. Energy from beam 17 is directed
to the reflectance accessory 11 in the form of a prism that
reflects or produces an incident beam 12 directed towards wafer 13
at an angle of incidence of less than 30.degree.. In a manner known
in the art, the epi layer on wafer 13 is transparent to the passage
of IR energy while the epi layer/substrate interface reflects such
energy. A portion of the incident energy is absorbed by the wafer.
Thus a portion of the energy from beam 12 is reflected as beam 14
which is then directed by the prism into an IR detector 18. This
detector produces an electrical output proportional to the
intensity of the IR energy received thereby which output represents
the intensity as a function of the path displacement of the mirror.
Such output plotted as a graph is known as an "interferogram". The
output of detector 18 is fed to a sample and hold (S and H) circuit
19 the output of which is fed into an analog-to-digital converter
(ADC) 20. In turn, the output of ADC 20 is fed into and stored
within the DPS 15.
Means are provided for controlling S and H 19 to digitize the
output of detector 18 at a predetermined sampling rate during a
scan. Such means includes a white light source 22 that directs
white light (polychromatic) through an interferometer 23 and into a
detector 24. Interferometer 23 contains a movable mirror (not
shown) linked to the movement of the mirror of interferometer 17.
In operation, the output of detector 24 produces a center burst
that is detected by a peak detector 25 to actuate a trigger 26 to
start sampling the IR or measuring beam. A laser source 30 directs
a monochromatic beam through an interferometer 31 whose mirror (not
shown) is also linked with the motion of the other mirrors, and
into a detector 32 whose output is a cosinusoidal function of the
mirror displacement. All mirrors are moved at a constant velocity
and the output of the detector 32 is fed through gate 33 so that
when trigger 26 has been actuated, gate 33 is opened to pass
signals from detector 32 into a Schmidt trigger 36. This component
functions to square the wave form from detector 32. The output of
trigger 36 is fed to a counter 37 whose output in turn is inputted
into a programmable divider 38. A cable 39 connected to DPS 15
delivers a control signal to set divider 38 to control the sampling
rate as a function of the number of laser signal cycles. Such
number is proportional to the mirror displacement of interferometer
31 between successive samples. Because the two interferometer
mirrors are moved together, the signal representing the intensity
of IR energy received by detector 18 is digitized to create a
series of digital values stored in DPS 15 as a function of mirror
displacement. Known programs stored in DPS 15 perform various
mathematical functions, including a discrete Fourier transform, to
convert such signals into digital values in the frequency domain.
Such values are the "spectral intensity", ie, intensity as a
function of wavelength.
"Spectral reflectance" is the ratio of the reflected intensity to
the incident intensity at each of the wavelengths across a spectral
band. To determine the incident intensity a platinum mirror is
substituted in spectrometer 10 for wafer 13. Such mirror has the
property of non-absorbtion of IR energy. The intensity of the light
reflected from the platinum mirror is the incident intensity which
is detected, digitized and stored. Spectral reflectance of the
wafer is then calculated by dividing the intensity of light
reflected from the wafer by the intensity of light reflected from
the platinum mirror, at each of the wavelengths.
DPS 15 includes a data library in which data measured at different
times can be stored and used in later calculations and
manipulations. Normally, the platinum mirror would be used once a
day to measure the "incident intensity", while many measurements of
reflected intensities from different wafers could be made and
spectral reflectances thereof calculated using the single
measurement of "incident intensity".
All of the foregoing apparatus and operations are the same as or
similar to those of the prior art. In summary, they perform step 40
(FIG. 2) of the process of the invention. The novel portions of the
invention are best understood from the remaining steps of the
method of the invention described below relative to FIG. 2.
Step 42 involves calculating the theoretical spectral reflectance
values for different epi thicknesses at least over a range
inclusive of the thickness being measured and dependent upon the
doping concentrations of the epi layer and substrate. The
calculations are based upon principles of classical physical optics
and are derived from two different theories previously
published.
The first theory is described in detail in Optical Properties Of
Thin Solid Films, by O. S. Heavens, Butterworths (1955) pages
76-77. In accordance with this theory, which deals with the optical
properties of a single absorbing layer on an absorbing substrate,
the reflectance R.sub.1 at a given wavelength and for a normal
angle of incidence, is as follows: ##EQU1## where n.sub.1 and
n.sub.2 are the indices of refraction of the epi layer and the
substrate, respectively, n.sub.0 is the index of refraction of air,
k.sub.1 and k.sub.2 are the coefficients of absorption of the epi
layer and the substrate respectively and d.sub.1 is the thickness
of the epi layer.
Referring to FIG. 3, a wafer 47 comprises an epi layer 48 grown on
a substrate 49 each of which is characterized by a different doping
concentration. When an incident ray 50 of a given wavelength is
directed onto the wafer, a first portion 51 is reflected from the
upper surface of epi layer 48. Another portion for ray 52 is
refracted downwardly through layer 48 and is reflected from the
interface, at point 56, between the epi layer and substrate, to
form a ray 53 which passes upwardly through the epi layer and exits
or forms a ray 54. Rays 51 and 54 interfere with one another in
accordance with classical interference theory to a degree dependent
in part upon the phase shifts that occurs at points 55 and 56.
This published theory dealt generally with the situation of any
absorbing layer on an absorbing substrate and we have found that
such general theory is applicable to the measurement to the
thickness of an epi layer. The values of n.sub.1, n.sub.2, k.sub.1
and k.sub.2 are dependent upon the doping concentration of the epi
layer and substrate and so in order to accurately calculate
reflectance by the above formulas, further calculations are done in
accordance with a second theory published in the article
"Measurement of Silicon Epitaxial Layers Less Than 1 Microbe Thick
by Infrared Interference", by P. A. Schumann, Jr., and C. P.
Schneider, Journal of Applied Physics, Volume 41, Number 8, July
1970, pages 3532-3535. This article contains formulae for
calculating n.sub.2 and k.sub.2 and the same formulae can be used
for calculating the n.sub.1 and k.sub.1 associated with the epi
layer. Such values are calculated as follows: ##EQU2## where N is
the carrier concentration, .rho..sub.0 is the DC resistivity,
.lambda. is the wavelength in a vacuum, m* is the conductivity
effective mass, e is the charge on an electron, C is the velocity
of light, K.sub.L is the dielectric constant of the entrinsic
semiconductor at long wavelengths and .GAMMA. is the gamma
function.
The calculations of theoretical spectral reflectance values are
performed preferably by forming a table or matrix of m rows by
(2q+1) columns, where m is the number of different wavelengths at
which the values are determined and q is a number large enough to
establish a range of different thicknesses including the one being
measured. The thicknesses at which the calculations are made is
determined by assuming a central value or nominal thickness and a
number q of different successive thicknesses differing by a fixed
amount h chosen according to the desired degree of precision and
the range of measurement. In a typical system; m=125 wavelengths
equally spaced across the IR spectrum of 2.5 to 50 microns; q=5;
and h=0.1 micron for d in the range 0.2 to 1.2 microns.
Step 44 involves a correlation analysis in which the measured
spectral reflectance values from step 40 are correlated with the
theoretical reflectance values of the different epi thicknesses
from step 42. This is done by calculating a series of correlation
coefficients between the theoretical spectral reflectance values
and the measured spectral reflectance values. There will be one
correlation coefficient for each of the assumed theoretical
thickness values. The correlation step assigns a numerical measure
to the degree of linear correlation between the theoretical and
measured quantities and in essence provides a quantitative measure
of how similar the theoretical and measured quantities are in shape
but not in absolute value. The calculations are done in accordance
with the following equation: ##EQU3## where CF is a matrix of 2q+1
columns of m rows of theoretical spectral reflectance values,
COR.sub.i is the correlation coefficient of the i th. column
corresponding to the i th. theoretical thickness value, CF.sub.ij
is the j th. element of the i column, CF.sub.i is the arithmetic
mean of the values in the i column, R is a vector of m values of
measured spectral reflectance, R is the arithmetic mean of such
value, and R.sub.j is the j element of vector R.
Each correlation coefficient is independent of amplitude
differences and can be any value between -1 and +1. If COR.sub.i
=+1 or -1, respectively, the relation is directly or indirectly
linear. If COR.sub.i =0, then there is no relation.
The terms eqn. 1-eqn. 18 are hereby defined to mean the equations
Eqn. 1 to Eqn. 18 as set forth above using the various values for
the letters and symbols described above. This definition applies to
the use of such terms hereinafter in the specification and in the
claims.
Step 46 then determines the epi thickness from the results of the
correlation analysis from step 44. This determination is done by
curve fitting the correlation coefficients to a quadratic fit,
locating the maximum positive peak of the fitted curve, and
interpolating the associated thickness values to determine the
thickness of the sample. By way of example and with reference to
FIG. 7, described in more detail below, the enlarged smooth curve
at the right of FIG. 7 represents the quadratic curve fitted around
the three greatest positive correlation coefficients. The peak is
at point 60 and the correspondiing interpolated value for thickness
is 0.28 microns.
Exemplary measurements of two different samples having different
epi thicknesses will now be discussed. Both samples are of silicon
wafers in which the epi layer carrier concentration is
1.times.10.sup.16 atoms of arsenic per cc and the substrate
concentration is 5.times.10.sup.19. The epi thickness of the first
sample measured or determined by the apparatus and method described
above was 0.51 microns. The measured spectral reflectance,
determined by step 40, is shown in FIG. 4. These measured values
were then correlated against theoretical values, determined by step
42, for epi thicknesses in the range 0.2 to 1.2 microns at eleven
equally spaced (0.1 microns) values to produce a correlation vector
whose values are plotted and shown in FIG. 5. A quadratic fit was
made of the values, the peak of the fit was located resulting in a
thickness of 0.51 microns. This result is the measured epic
thickness of the first sample.
FIGS. 6 and 7 are graphs, similar to FIGS. 4 and 5, of the spectral
reflectance and correlation of the second sample for which the
measured thickness was determined to be 0.28 microns. These two
sample measurements are for illustrative purposes only and are not
to be considered as limiting. The lower limit or ranges of epi
thickness for the invention is less than 0.1 micron and the upper
limit is above that value for which semiconductor manufacturers
currently have an interest and greater than 35 microns.
It should be noted that the incident beam 12 in FIG. 1 is not
normal or perpendicular to the wafer, while the first theory (eqns.
1-17) assumed a normal angle of incidence. This difference results
in an error of less than 0.01 microns and can be considered
negligible.
The preferred embodiment of the invention involves programming a
general purpose computer by storing a program therein so that the
computer and stored program constitute the apparatus form of the
invention. Such apparatus automatically performs the functions or
process described above. The details of the program are not germane
to understanding the invention and it is merely a clerical
programming skill to translate the method described above,
including performing the various mathematical algorithms, into a
suitable program.
The invention is also susceptible to many different variations.
First, the above description implies creating a single table or
matrix encompassing the thickness of a sample. Where samples of
greater ranges of thicknesses and doping concentrations are to be
measured, several tables or matrices can be created and stored. By
known procedures, such tables can be individually accessed during
the measurement of a single wafer. This general procedure thus
speeds up measuring many wafers particularly in a production
environment. Secondly, a slightly different form of the invention
was published by the assignee of the invention in the IBM TDB Vol.
24, No. 1A (June 1981), pages 49-54.
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